[aliromman]

This constructs the inverse of the eight-mode coupled provider in PR #5402. All 632 original input directions are retained and all nine owned fields return zero. The paid profile remains {1:2193,2:412,3:510,4:127,5:40}: no extra Gaussian child. See the proof and local bill and exact receipt.

The recipe removes the unused C0 accumulator, reverses the remaining native operations, and uncomputes the eight retained copies by subtraction. It reverses the physical boundaries with their exact phases. Implementing inverse children as positive Gaussian calls costs at most 3,282 additional ordinary bit translations; these are included in the local bill. Only research/native-coupled-inverse/ is added.

This is an enabling construction, with no new kappa. The eight mode outputs remain coupled. The complete caller, its precision/compiler costs and outer admission remain open. The all-width result inherits the pinned source's native implementation model. An inverse also reverses its D factor; it does not provide a fresh independent D factor by itself. No selected-frontier change is made.

Validation passed under Python 3.12.13:

• Registered normal and isolated optimized replay: both compositions on all 632 original columns, nine zero owned returns, separate inverse formula, exact chart and paid-count checks, 464 boundary adjoints and inverse-child phases for ranks 1 through 9.
• The frozen predecessor replay: all 641 columns and 1,240 physical phase bindings. Three new controls reject inverse COPY sign, zero-retag input and boundary phase errors.
• All eight archived predecessor files match their exact source Git blobs. Final committed-byte checks cover 16 package/input files. Official export integrity and a fresh read-only Git-free optimized replay pass.

python3.12 -B scripts/research_package.py --package research/native-coupled-inverse
python3.12 -I -O -B research/native-coupled-inverse/verify.py

Expected final token: PASS_SAME_PAID_NATIVE_TWO_SIDED_INVERSE. The validation record states the limits. The optional full-history ancestry preflight stopped at its shallow-checkout guard and did not run. Hosted planning and registered replay passed at head 123ffa6aa7c1866ac60c67739841f71d95b3edf7. Independent human review, a selected cold replay, full recursive compiler/outer admission and formal-kernel checking are not claimed.

Base: canonical main 1ec1119972c9aefe42525172d07b332fcff81cb4, priority revision 7. The source is fixed at PR5402 head 3348fccea855082786d5f0fc5558b73d4974e745; its exact files are retained for offline replay with origins and byte hashes. Canonical main and live PR/Discussion overlap were refreshed before submission. The caller claim continues separately.

Credit Dillon for the coupled provider and physical basis in PR #5402, and Ali Romman for the cached-center identities in PR #5383. Yiyang's PR #4966 already gives owned-center inverses for the original word by recomputation; this result supplies the inverse for the changed retained-cache interface. Cao Yifan, Dugongue, Cinna Davis, zxv, Jacob Alan Sussman, Ryanson Jonathan and the full source lineage retain their credit. Prepared for Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.